If you do not assume prior to your investigation that the amount of CO2 as a function of time is most probably exponential, then, well, you need to do both fits, linear and exponential (and preferably other parametrizations) and decide which fit is “better” (since this post is not a chapter in a textbook I won’t define what is “better”, take any sensible definition from statistics). If you do this, you will find that the best linear and best exponential fits are almost indistinguishable with very similar chi^2/dof values.

See the result of the above analysis where I performed both linear and exponential fits. If you are an honest statistician, you can not possibly claim that the exponential fit is in any way “better” than the linear fit. They are actually almost indistinguishable. The chi^2/dof values are large anyway and not much different, 138.1 for the linear fit and 94.5 for the exponential. The difference is not large and, again, both are very far from 1.

If the pre-industrial value of 280 ppmv is subtracted from the data the fitting can be repeated. Obviously there will be no change to the linear fit but the exponential fit will do change. Actually, the new chi^2/dof value will go down to 45.9 but that is still huge and very far from 1: comparison of these two fits.

In conclusion, the data set does not allow for a conclusion that “global CO2 levels are rising exponentially and not linearly”, an honest assessment would need to reject both proposals or if the requirements are relaxed, conclude that the data set can not meaningfully distinguish between the two parametrizations.

The reason is actually quite obvious: the data set contains too few data points.